Numerical Solution of Differential Equations in Mechanics

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9870399 ArnoldThe investigator will devise, improve, and analyze methods for the numerical simulation of complex physical phenomena modeled by partial differential equations. Three major areas of application will be studied, each requiring special methods: computational relativity, certain important classes of differential equations arising from incompressible fluid flow and from electromagnetism, and linearly elastic plates.In the area of computational relativity, the work is motivated by the search for gravitational radiation using the LIGO detector, presently under construction. LIGO is the prime example of a new generation of gravitational wave detectors, and is being designed to detect gravity waves (which have hitherto escaped detection due to their small amplitude). If the detected wave can be analyzed to determine the nature of the massive but distant astronomical event which gave birth to it (e.g., the spiraling coalescence of a pair of black holes), then LIGO will become mankind's first window on the universe that can "see" beyond the electromagnetic spectrum (light and radio signals). The proposed project is to devise efficient algorithms to enable such analyses to be performed on high performance computers. These algorithms are based on Einstein's equations underlying general relativity. The investigator will both extend and apply already completed computer codes for the constraint equations, purely spatial partial differential equations arising from the Einstein equations, and will also study the problem of numerically solving the Einstein evolution equations. This is highly interdisciplinary work, involving extensive collaboration between the investigator, a mathematician, and his group, and physicists and astronomers.The second area of study concerns the developments of multigrid algorithms and supporting theory for solving certain sorts of differential equations which have hitherto resisted such approaches. Multigrid algorithms, which exploit a decomposition of the desired solution into components on many different scales, are among the most efficient algorithms invented for solving many sorts of differential equations. However for an important class of problems arising from incompressible continua and another group of problems arising from Maxwell's equations of electromagnetism, new multigrid approaches are required.Finally the investigator will collaborate on a monograph on the theory of elastic plates, including modeling, analysis, and numerical simulation. There has been extensive progress in these areas over the past decade, and the book will serve as both a reference for workers in the field and in the active field of elastic shell modeling, and as an introduction for researchers new to the area.